Stochastic methods for simulating uncertainties in free stream turbulence and in the geometry Project MUNA: Final Workshop, Alexander Litvinenko, Institut f ¨ ur Wissenschaftliches Rechnen, TU Braunschweig 0531-391-3008, [email protected]March 22, 2010
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Stochastic methods for uncertainty quantification in numerical aerodynamics
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Stochastic methods for simulatinguncertainties in free stream turbulence and in
the geometry
Project MUNA: Final Workshop,Alexander Litvinenko,
Figure: [min, max] intervals in each point of RAE2822 airfoil for the cp and cf.The data are obtained from 645 solutions, computed in n = 645 nodes ofsparse Gauss-hermite grid.
Figure: Intervals [mean− σ, mean + σ], σ standard deviation, in each point ofRAE2822 airfoil for the pressure, density, cp and cf. Build for 645 points ofsparse Gauss-Hermite grid.
α(θ1, θ2), Ma(θ1, θ2), where θ1, θ2 have Gaussian distributions
Table: Statistics obtained on sparse Gauss-Hermite grid with 137 points.
PCE of order 1 with 3 random variables and sparse Gauss-Hermitegrid wite 25 points were used.
Outline
Overview
Modelling of free stream turbulenceNumerics
Uncertainties in geometryNumerics
Low-rank approximation of the solutionNumerics
Low-rank approximation of the solution
U VΣ T=M
UVΣ∼
∼ ∼ T
=M∼
Figure: Reduced SVD, only k biggest singular values are taken.
Decay of eigenvalues
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−20
−15
−10
−5
0
5
log, #eigenvalues
log
, va
lue
s
pressuredensitycpcf
Figure: Decay (in log-scales) of 100 largest eigenvalues of the combinedmatrix constructed from 645 solutions (pressure, density, cf, cp) on thesurface of RAE-2822 airfoil.
1. A.Litvinenko, H. G. Matthies, Sparse Data Representation ofRandom Fields, PAMM, 2009.
2. B.N. Khoromskij, A.Litvinenko, H. G. Matthies, Application ofhierarchical matrices for computing the Karhunen-Loeveexpansion, Springer, Computing, 84:49-67, 2009.
3. B.N. Khoromskij, A.Litvinenko, Data Sparse Computation of theKarhunen-Loeve Expansion, AIP Conference Proceedings,1048-1, pp. 311-314, 2008.
4. H. G. Matthies, Uncertainty Quantification with Stochastic FiniteElements, Encyclopedia of Computational Mechanics, Wiley,2007.
Acknowledgement
Elmar Zander
A Malab/Octave toolbox for stochastic Galerkin methods(KLE, PCE, sparse grids, tensors, many examples etc)